Systems and methods herein generally relate to the problem of calculating powerflow studies of electrical networks, and more particularly to methods for incorporating and simulating the behavior of power system controls in those studies.
Powerflow studies need to incorporate the effects of several kinds of regulating devices that are always present in the operation of electrical networks. A non-exhaustive list of the most interesting ones, from the point of view of steady-state powerflow, comprises: voltage regulation by generators (AVR, Automatic Voltage Regulation) or by transformers (ULTC, Under-Load Tap Changers), real power regulation by phase shifters, reactive power regulation by transformers, and net real power regulation across tie lines (area interchange schedules). Additionally, controls may be local or remote. Some remote controls may be real, enabled by modern fast telecommunications; others are just convenient artifacts used in the context of planning studies. Whatever the case, a power flow method needs to incorporate all these types of control in order to be useful for real work.
Iterative powerflows use one of these two different general approaches: (a) integrating the additional equations and new control variables into the definition of the matrix of the method (possibly eliminating some variables if they are given directly by the control setpoints); (b) keeping the original equations and using an “outer loop” approach, whereby the control variables are adjusted in between iterations, in proportion to the residuals of the regulated magnitude. The proportionality coefficients for these adjustments, the so-called sensitivities, are obtained either by theoretical modeling, direct computation, or empirical tests. This option (b) is favored by methods that keep the Jacobian constant through the iterations, such as the FDLF method of Stott and Alsac, because it allows taking into account control limits on the fly. This, however, makes convergence behavior even harder to model and analyze. By contrast, the treatment of limits in method (a) requires a change in the equations (for instance, a PV to PQ type switch), so it would be more suited for a full NR method. Apart from the inherent problems of iterative load flow methods, the problem with both of these two approaches to the treatment of controls is that the unpredictable dynamics of the numerical iteration is never a good guide to select a solution when there are many possible saturated controls (as it is often the case in planning studies).
U.S. Pat. Nos. 7,519,506 and 7,979,239 to Trias take a very different approach. The method, from here onwards termed the Holomorphic Embedding Load-flow Method (HELM), is non-iterative, constructive, and takes advantage of the specific mathematical structure of the power flow problem by using techniques of Complex Analysis. However, the method, as disclosed there, focused on the fundamentals of the load flow calculation and only gave explicit implementation details for PQ buses, with no regulating devices. The innovation disclosed here extends the aforementioned method in order to contemplate control devices and correctly calculate the steady state solution of the network when they operate, all while preserving the nice deterministic properties of the base method.